By now considering the random walk operator as a reference operator and its associated eigenvectors as a Fourier basis for functions defined on directed graphs, the last part of our harmonic analysis on directed graphs consists in building a multi-resolution analysis for functions defined on the vertices of an directed graph. Complex graphs, both directed and undirected, have structures at different scales. Motivated by the efficiency of multi-resolution wavelet analysis in traditional signal processing [1]} and the multi-scale dimension of graph data, a number of multi-resolution graph constructions have emerged in recent years [2]}, [3]}, [4]}, [5]}, [6]}. Here, we propose a new harmonic analysis built around the random walk operator and whose multi-resolution constructions generalize graph spectral wavelets [4]}, [3]} and diffusion wavelets [2]}, [10]} on directed graphs.
